How To Replace External Hard Drive So System Finds Files After Switch

I need to replace an external hard drive. What is the best way to go about this so when I am done the OS and individual pieces of software finds the files automatically. In the past I tried naming the new drive the same as the old drive but that did not help and I basically had to go into every piece of software and tell it that the file it is looking for that used to be on "External Drive X" is still on "External Drive X."

Is there a way to make this happen more efficiently? I cannot find any sort of tutorial or help topic to assist.